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import OAI.Geometry.HyperbolicGroups.DevelopmentGeometricAction
import OAI.Geometry.HyperbolicGroups.LocalGeodesic
import OAI.Geometry.HyperbolicGroups.GeometricSpaceTransport
namespace OAI
section
/-! Compact locally CAT(-1) geodesic metrics yield genuine geometric CAT(0)
actions of the actual fundamental group. -/
noncomputable section
namespace HyperbolicObstruction
open Set Metric
open scoped Topology ContinuousMap
universe u
lemma not_noGeometricCATZeroAction_of_compact_localCATNegOne
{X : Type} [MetricSpace X] [CompactSpace X] (o : X) (g : IsGeodesic X)
(hl : ∀ x : X, ∃ r : ℝ, 0<r ∧ CATNegOne (ball x r)) :
¬ NoGeometricCATZeroAction.{u} (FundamentalGroup X o) := by
intro h
obtain ⟨r,hr,hc⟩:=uniformLocalCATZero_of_compact hl
let e := LocalGeodesic.deckPiOneEquiv (o:=o) hr hc g
let A := LocalGeodesic.deckGeometricAction (o:=o) hr hc g
let := LocalGeodesic.intrinsicMetric (o:=o) hr hc g
let := LocalGeodesic.developed_proper (o:=o) hr hc g
let : Nonempty (LocalGeodesic o) := ⟨LocalGeodesic.constant o⟩
let Y := ULift.{u} (LocalGeodesic o)
let : ProperSpace Y := ulift_proper
have hcat : CATZero Y := CATZero.of_isometryEquiv (uliftIsometry (LocalGeodesic o))
(LocalGeodesic.developed_catZero hr hc g)
have : IsEmpty (GeometricAction (FundamentalGroup X o) Y) :=
h Y inferInstance inferInstance inferInstance inferInstance hcat
exact isEmptyElim ((A.pullback e).spaceTransport (uliftIsometry (LocalGeodesic o)))
lemma incompatibleLocalCATNegOne_of_noAction (L : FiniteComplex) (y : L.Carrier)
(h : NoGeometricCATZeroAction.{u} (FundamentalGroup L.Carrier y)) :
¬ CompatibleLocallyCATNegOne L := by
rintro ⟨m,hm,hg,hl⟩
have hcompact : CompactSpace L.Carrier :=
inferInstanceAs (CompactSpace (FinitePolyhedron.Carrier L.complex))
let P : TopologicalSpace L.Carrier → Prop := fun t =>
letI := t
NoGeometricCATZeroAction.{u} (FundamentalGroup L.Carrier y)
let := m
let : TopologicalSpace L.Carrier := m.toUniformSpace.toTopologicalSpace
let : CompactSpace L.Carrier := by
change @CompactSpace L.Carrier m.toUniformSpace.toTopologicalSpace
rw [hm]
exact hcompact
apply not_noGeometricCATZeroAction_of_compact_localCATNegOne y hg hl
exact (congrArg P hm).mpr h
lemma homotopy_incompatibleLocalCATNegOne {K L : FiniteComplex} (x : K.Carrier)
(h : NoGeometricCATZeroAction.{u} (FundamentalGroup K.Carrier x))
(e : K.Carrier ≃ₕ L.Carrier) : ¬ CompatibleLocallyCATNegOne L :=
incompatibleLocalCATNegOne_of_noAction L (e x)
(h.of_mulEquiv (fundamentalGroupHomotopyEquiv e x))
end HyperbolicObstruction
end
end
section
end
end OAI
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